Elastic Properties of Solids Topics Discussed in Kittel, Ch. 3, pages 73-85. Hooke's “Law”.
A property of an ideal spring of spring constant k is that it takes twice as much force to stretch the spring twice as far. That is, if it is stretched a distance x, the restoring force is given by F = - kx. The spring is then said to obey
A Mass-Spring System
in which a mass m is
attached to an ideal
spring of spring
constant k. That is, the
Stretch the spring a distance A & release it:
∑Fx= 0 = mg - kx0
or x0 = (mg/k)
Newton’s 2nd Law
Equation of Motion:
This is the same as
before, but the
equilibrium position is
x0instead of x = 0
Simple Harmonic Oscillator
Hooke’s “Law” for a vertical spring (take + x as down):
An Elastic Medium is defined to be one in which a disturbance from equilibrium obeys Hooke’s “Law”so that a local deformation is proportional to an applied force.
If the applied force gets too large, Hooke’s “Law” no longer holds. If that happens the medium is no longer elastic. This is called the Elastic Limit.
The Elastic Limit is the point at which permanent deformation occurs, that is, if the force is taken off the medium, it will not return to its original size and shape.
theycorrespond to the low frequency, long
wavelength limitof the more general lattice
vibrations we have been considering up to now.
Consider Longitudinal Elastic Wave
Propagation in a Solid Bar
So, consider sound waves propagating in a solid, when theirwavelength is very long, so that the solid may be treated as a continous medium. Such waves are referred to as elastic waves.
In general, a wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The Stress Sat a point in space is defined as the force per unit area at that point.
To wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The analyze the dynamics of the bar, choose an arbitrary segment of length dx as shown above. UseNewton’s 2nd Law to write for the motion of this segment,
Mass Acceleration = Net Force resulting from stress
Equation of wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The Motion
So, this becomes:
Cancelling common terms inAdx gives:
This is the wave equation a plane
wave solution which gives the
Plane wave solution:
k = wave number = (2π/λ), ω = frequency,A = amplitude
Unlike the case for the discrete lattice, the wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The dispersion relation ω(k)in this long wavelength limitis the simple equation:
The speed VL with which a longitudinal elastic wave moves through a medium of density ρis given by:
C Bulk Modulus
ρ Mass Density
Speed of wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The Sound for Several Common Solids
Most calculated VL values are in reasonable agreement with measurements. Sound speeds are of the order of 5000 m/s in typical metallic, covalent & ionic solids :